The subsampling bootstrap and the moving blocks bootstrap provide effective methods for nonparametric inference with weakly dependent data. Both are based on the notion of resampling (overlapping) blocks of successive observations from a data sample: in the former single blocks are sampled, while the latter splices together random blocks to yield bootstrap series of the same length as the original data sample. Here we discuss a general theory for block bootstrap distribution estimation for sample quantiles, under mild strong mixing assumptions. A hybrid between subsampling and the moving blocks bootstrap is shown to give theoretical benefits, and startling improvements in accuracy in distribution estimation in important practical settings. An intuitive procedure for empirical selection of the optimal number of blocks and their length is proposed. The conclusion that bootstrap samples should be of smaller size than the original data sample has significant implications for computational efficiency and scalability of bootstrap methodologies in dependent data settings. This is joint work with Todd Kuffner and Stephen Lee and is described at https://arxiv.org/abs/1710.02537.